91Ó°ÊÓ

Suppose that \(f(x)=e^{-|x|}, x \in[-2,2]\). (a) Show that \(f(-2)=f(2)\). (b) Compute \(f^{\prime}(x)\), where defined. (c) Show that there is no number \(c \in(-2,2)\) such that \(f^{\prime}(c)=0\). (d) Explain why your results in (a) and (c) do not contradict Rolle's theorem. (e) Use a graphing calculator to sketch the graph of \(f(x)\).

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 1} \frac{x^{2}-1}{x+1} $$

Find the general solution of the differential equation. $$ \frac{d y}{d s}=\cos (2 \pi s), 0 \leq s \leq 1 $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0} \frac{1-\cos x}{x} $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow-\infty} x e^{x} $$

Find the general solution of the differential equation. $$ \frac{d y}{d x}=\frac{1}{1-x}, x>1 $$

Find the general solution of the differential equation. $$ \frac{d y}{d x}=\frac{1}{x+1}, x>-1 $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0^{+}}\left(\frac{1}{x}-\frac{1}{\sqrt{x}}\right) $$

Solve the initial-value problem. $$ \frac{d y}{d x}=3 x^{2}, \text { for } x \geq 0 \text { with } y(0)=1 $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow+\infty} \frac{\sqrt{x+1}}{\sqrt{x}} $$